CRI has "meaning" at 3000K and below, sort of, maybe, maybe up to 3200K, and at 5000K and above. Below 5000K, CRI is purely reference to how some color samples "look" compared to how they would like if illuminated with a black body radiator, i.e. a hot filament bulb. Few have any experience at all of a hot filament bulb above 3000K for illumination, as that is the highest typical for halogen bulbs, though some automotive run hotter, but that is not general illumination. Over 5000K and it is some agreed upon version of sunlight so we have a reference, sort of.
Simplest analog to the CRI. Mix a 3000K and 5000K LED in the right mix, and you will perceive 4000K. Mix a 2700K LED and a 6000K LED in the right mix and you will also perceive 4000K. The spectrums of each 4000K equivalent will be different.
4000K is what your eyes perceive when the light is shined into them. CRI is what your eyes perceive when that same light is used to illuminate objects with light of equivalent CCT. An LED/OLED screen can produce almost any perceived color of light with RGB, but when you use that light to shine on things, it looks much different from a white light of the same CCT.